Wakayama, JAPAN

　

　

A theoretical model is introduced for realizing a coastal upheaval by an ocean water loading. An increase of the ocean water loading can be expected by the resultant effect of the global warming, which might be caused by the climatological warming and glacial melting which must be resulted by the recent human activities after utilizing the fossil fuels and with producing the artificial chemical products which affect strongly the earth's environment. This model is a simple linearized elastic model of a thin plate assumed to be equivalent to the existing tectonic plate on the earth. This model plate is for demonstrating and evaluating upheaval just around the coastal zone. A set of the differential equations for an equivalent elastic thin plate is solved with consideration of several equivalent parameters specifying the elastic constants. The solution under some assumed conditions suggest that the ocean water loading is effective to give a set back of the coastline. In this work, an additional brief note on time factor is given though the model is for an upheaval pattern. The solution might be well applied for some other geophysical problems, for example, on dynamical understanding of an existing profile of geographic section and on a dynamical mechanism to realize what process is possible at forming a co-seismic upheaval pattern.

　

　

A theoretical model is introduced for realizing a coastal upheaval by ocean water loading. This model might give us a key to get a dynamical understanding of what could be expected as a final stage of the threat of an increasing sea level caused by ocean water loading. This ocean water loading is considered to be caused mainly by resultant effect of the global warming. The global warming is essentially one of the important factors of climatology. The climatological process of the global warming must be controlled by the natural variations of the solar beam on the earth's surface and by the artificial impacts of the man-made products appeared and increased after the recent human activities after utilizing the fossil fuels and finding the industrial chemical products. It is now widely known that these impacts are resulted to the growth of the ozone layer in the polar upper layer by the artificial effect. Then, we have to consider here these products, which might surely be found in the ocean in near future. Several products are discussed in a scope of environmental problems in relation to the factors, for example, carbon dioxide but CFC (Chrolo-Fluoro-Carbonate) or NOx, SOx, and the other fine particulate materials. Nevertheless, in this work, simply a dynamical problem is considered for considering the final stage of the global warming. The term of this global warming might concerns directly to the climatological problem, though the author's interest is mainly in the problem of the crustal upheaval caused by the global warming.

　

In this work, a linearized model of the author's interest is introduced first for an elastic thin plate that could be taken as an equivalent crust. This crust is considered to be a part of the existing tectonic plate on the earth. This model plate is for demonstrating and evaluating crustal upheaval just around the coastal zone. A set of the differential equations for the equivalent elastic thin plate is introduced to formulate the interested problem in this work with considering several equivalent parameters specifying the elastic constants. The solution under some assumed condition suggests that the ocean water loading is effective to give a set back of the coastline. The solution might be well applied for some other geophysical problems, for example, on a dynamical understanding of an existing profile of a geographic section in the coastal zone, and on a key to a dynamical mechanism of a forming process of a co-seismic upheaval pattern. In this work, a brief note on time factor is given.

　

　

Looking at the natural topography in the coastal zone, it can be seen a mountain range along the coast line with a continental shelf and a trough between the coast line and the deep sea area in several locations of the coastal zone around the existing continents.

　

This natural pattern of the topography in the coastal zone stimulates us the natural scientists to raise a problem for geographic or other related scientific fields. In fact, it is hard to consider the time-span of any process of the topographic formation by using a simple spatial model. There is a long history of the researches in the past in the related fields (i.e., geography, geology, geomorphology, geodesy; geophysics, and others), so that it would be hard to introduce and list up here all of the contributions. These contributions should be referred in the other publications, and no list is shown except what are directly referred in this work.

　

Regarding the mathematical theory of the elastic earth, Jeffreys (1952) has published his classic theory after Love's (1927) treatise on the mathematical theory of elasticity.

　

Regarding the process of crustal upheaval, it is necessary to take a geological time scale rather than any other time scales utilized in the existing fields of, for example, meteorology, climatology, oceanography, seismology, and etc. There is no current dynamical research available on the time-span of the process of crustal upheaval. We merely understand that crustal motion and its resulting change take place very slowly. Research on plate tectonics includes, for example, Care (1976) on scale of tectonic phenomena, as well as Davies (1999) and Richard et al. (2000).

　

Recent climatological studies have also focused on global trends (see, for example, Clark et al., 1994, 2000) on the basis of a two-dimensional ocean-climate model considering the thermal effect of atmospheric temperature increase on the seawater column in the world. Watt (1977) also noted sea level rise. Singer (1999) and Nakamura (1999) introduced some notes in relation to what had been reported in Houghton's publication on IPCC (1990).

　

Once, Nakamura (2001) has studied on crustal upheaval in a coastal zone in relation to an expected increased loading of the oceanic water on the crust after global warming and glacial melting. A comment in a public brochure motions that a future crustal rebound, as a result of an unloading effect of a melting glacier, is possible. Thus, future study of such a reversal possibility of crustal upheaval is warranted.

　

　

Assuming an elastic plate in a coastal zone instead of a shell covering the earth, a crustal upheaval can be taken as a problem of the elastic plate. The thickness of the crust is less than one hundredth of the earth's radius. So then, this assumption is reasonable.

　

For convenience, the plate is assumed to be of uniform and homogeneous (thickness. H, and density,ρ). Then, it becomes easy to formulate the problem for this study. Formulating technique for this purpose is essentially same as that for the classic problem on elastic bodies (see, for example, Love, 1892; Jeffreys, 1852; Officer, 1974; Nakamura, 2001). The surface of the plate, then, is taken as the horizontal reference (O-x in Fig.1) with a static loading of the semi-infinite oceanic layer (thickness Ho and density ρ₀ on the plate.

　

When a quasi-equilibrium state of the crust is assumed, the crustal upheaval can be obtained by solving a formulated equation such as that for the classic problem of elastic bodies. The vertical displacement (w) of the crust surface by loading is reduced from the equations of motions for the elastic body with some assumptions. In this case, the reduced equation is written as follows:

　

　

Where,

　

　

And D(flexural rigidity of the plate) is

　

　

where, E is the Young modulus, and σ is the Poisson ratio. Introducing Lame's two constants, λandμ, the two parameters, E and σ are written as (cf. Officer, 1974; Nakamura, 2001):

　

　

And,